{"id":"6fe0a9d7-27ff-4df2-9585-f7fee85b6a41","arxiv_id":"2510.13309","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Canonical actions of Thompson's group V and topological full groups of amenable ample groupoids on the Cantor set are not strongly ergodic, so their crossed products are non-full factors (type III_{1/d} for Higman–Thompson).","lead":"The paper proves that the canonical actions of Thompson's group V and its generalizations (Higman–Thompson, Brin–Thompson, and topological full groups of amenable ample groupoids) on the Cantor set are not strongly ergodic. Consequently the associated crossed product von Neumann algebras are not full, even though the group von Neumann algebra of V is full.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.2 is false as stated: the finite two-point groupoid gives a strongly ergodic canonical Z/2 action; an aperiodicity/Cantor-unit-space hypothesis is missing.","rationale":"The reader's flagged weakest assumption was Lemma 3.1, the transfer from groupoid amenability to equivalence-relation amenability. That lemma is the least problematic step: the pushforward estimate is exactly the groupoid amenability condition, and the uniform convergence on compact subsets applies to the singleton {g}. The actual load-bearing flaw is the quantification in Theorem 3.2. The finite two-point groupoid satisfies every stated hypothesis and its canonical topological-full-group action is the transposition on two points, which is strongly ergodic. This is not merely a missing hypothesis in an ancillary note; it falsifies the headline theorem and the statement that no topological full group of an amenable ample groupoid admits a strongly ergodic quasi-invariant action. It also explains why the Rokhlin-lemma step cannot be applied without an aperiodicity assumption: finite equivalence relations do not admit non-trivial asymptotically invariant sequences for their full groups. The paper's main examples are unaffected — G_{d,k} has Cantor unit space and infinite orbits — so the correction is to add an explicit hypothesis (e.g., no isolated points / all orbits infinite) to Theorem 3.2 and Corollary 3.3, and to remove or qualify the general claim. I therefore keep the disposition CONDITIONAL but for a substantive reason rather than typographical issues, and I partially agree with the reader: the finite-action observation was already noted parenthetically, but its impact on Theorem 3.2 was missed.","tokens_in":9182,"tokens_out":21247,"duration_ms":172990,"concrete_test":"Apply Theorem 3.2 to the two-point groupoid G={id_a,id_b,α,α^{-1}} with α:a→b. Check each hypothesis: compact Hausdorff étale ample groupoid; topologically principal; amenable with the stated measures. Compute [[G]]≅Z/2 swapping a,b and verify strong ergodicity of this action for μ=(δ_a+δ_b)/2 as above. The contradiction is immediate and settles that the theorem needs an extra aperiodicity/perfectness assumption.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The universal statement of Theorem 3.2 is false. Let G be the finite groupoid with units a,b, identity arrows id_a,id_b, and one arrow α:a→b with inverse α^{-1}. It is locally compact Hausdorff with compact unit space, étale, ample (discrete topology is totally disconnected), topologically principal (all isotropy groups are trivial), and amenable: take m^a=δ_{α^{-1}} and m^b=δ_α, then α·m^a=m^b and α^{-1}·m^b=m^a, so the amenability net can be chosen exactly invariant. The topological full group [[G]] is {id, σ}, where σ swaps a and b. The uniform measure μ=(δ_a+δ_b)/2 is quasi-invariant, and the action σ↷{a,b} is strongly ergodic: if μ(A_n△σA_n)→0, then since μ values on {a,b} are atomic, eventually A_n is σ-invariant, hence A_n=∅ or {a,b}, so μ(A_n)(1-μ(A_n))→0. This directly contradicts Theorem 3.2. The proof fails in the invocation of Schmidt's Rokhlin lemma ([19, Prop. 2.2]): a finite equivalence relation has no non-trivial asymptotically invariant sequence for its full group. Lemma 3.1 (groupoid amenability ⇒ Borel amenability of R_G) appears sound; the gap is the missing hypothesis in Theorem 3.2 excluding finite orbits — e.g. requiring G^{(0)} to have no isolated points (a Cantor space) or all orbits to be infinite. The Higman-Thompson groupoids G_{d,k} satisfy this, so the concrete Thompson-group conclusions are not affected, but the theorem as printed is false.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a general theorem: for a topologically principal, amenable, ample groupoid G, the canonical action of its topological full group [[G]] on the unit space is claimed to be not strongly ergodic with respect to any quasi-invariant probability measure. The proof combines Lemma 3.1 (groupoid amenability implies amenability of the associated orbit equivalence relation R_G) with Schmidt's Rokhlin lemma to produce a non-trivial almost invariant sequence. As applications, the paper shows that the canonical actions of the Thompson, Higman–Thompson, and Brin–Thompson groups are not strongly ergodic, their associated crossed products are not full, and derives a non-embedding corollary for strongly ergodic actions. It then specialises to the Higman–Thompson group V_{d,k} and computes the crossed product to be a non-amenable, non-full factor of type III_{1/d}.","tokens_in":9440,"tokens_out":12756,"duration_ms":101993,"significance":"If the universal statement were correct, the paper would give a clean structural result: no topological full group of an amenable ample groupoid can admit a strongly ergodic quasi-invariant action on its unit space, and the accompanying non-embedding and non-fullness corollaries are natural and useful. The proof strategy is transparent and mostly standard: Lemma 3.1 gives a direct transfer from groupoid amenability to Borel amenability of R_G, and the subsequent ergodicity and type III computations are classical modulo minor typos. The paper is honest about the non-constructive nature of the almost invariant sequence. However, the central theorem as printed is false: a finite two-point groupoid is a counterexample. The fix is likely local and the Thompson-group applications survive, but the overstatement is load-bearing in the current version.","major_comments":[{"comment":"Theorem 3.2 is false as stated. Let G be the finite groupoid with unit space {a,b}, identity arrows id_a,id_b, and one arrow α:b→a with inverse α^{-1}. This groupoid is locally compact, Hausdorff, étale, ample, and topologically principal. It is amenable: for example, take m^a=δ_α and m^b=δ_{id_b}; then α·m^b=m^a and α^{-1}·m^a=m^b, so the amenability net in §2.1 is exactly invariant. The topological full group [[G]] is {id,σ}, where σ swaps a and b, and the uniform probability μ on {a,b} is quasi-invariant. The action σ↷({a,b},μ) is strongly ergodic: any almost invariant sequence is eventually σ-invariant, hence trivial. This directly contradicts the theorem. The missing hypothesis is an aperiodicity assumption (e.g., G^(0) has no isolated points, or every G-orbit is infinite, or μ gives zero mass to finite G-orbits). Please add such a hypothesis; the Thompson/Higman–Thompson groupoids","section":"Section 3, proof of Theorem 3.2"},{"comment":"The proof invokes [19, Proposition 2.2] to obtain, from μ-amenability of R_G, a non-trivial almost invariant sequence for the full group [R_G]. This is the exact step that fails for the finite groupoid counterexample described above. The citation's hypotheses must be checked explicitly: an amenable equivalence relation with finite classes does not, in general, admit such a sequence. Please either add the missing aperiodicity hypothesis to Theorem 3.2 and verify it before applying the Rokhlin lemma, or prove the existence of the almost invariant sequence directly under the stated hypotheses. Without this, the universal statement is unsupported even apart from the explicit counterexample.","section":"Section 3, proof of Theorem 3.2"}],"minor_comments":[{"comment":"In the final display of the proof, the norm should be ∥g m_i^{s(g)} − m_i^{r(g)}∥_1, not ∥g m_i^{s(g)} − m_i^{s(g)}∥_1 as printed. The preceding line makes the intended estimate clear, but it is a typo in a load-bearing inequality.","section":"Lemma 3.1"},{"comment":"The notation for R_x and R^x is confusing and likely misprinted: the text defines both R_x and R^y as {(y,x)∈R}. In Lemma 3.1, p_i^x is used as a measure on the range fibre, so the definition of R^x should be clarified (presumably {(x,y)∈R}). This would improve readability.","section":"Section 2.2"},{"comment":"The sentence 'and (L∞(X_{d,k}) ⋊ V_{d,k})^{σ^φ} is a factor' is stated without proof. Since this fact is needed to apply [6, Corollary 3.2.7] for the S-invariant computation, a brief justification (or a reference to a standard argument) would be helpful.","section":"Section 3, type III computation"}],"recommendation":"major_revision","confidential_remarks":"The reader's counterexample is correct and decisive: the universal formulation of Theorem 3.2 is false, and the proof of that theorem relies on a Rokhlin lemma whose hypotheses are not verified. That said, the mistake is a missing aperiodicity hypothesis, not a failure of the main mechanism or the Thompson-group applications. The paper should be reconsidered after a revision that adds the missing hypothesis and verifies it for the groupoids of interest. The concrete results for V_{d,k} appear sound modulo the standard typo-level issues noted in the report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's concrete result on Thompson-type groups is likely correct, but the headline theorem as stated is false: the finite two-point groupoid with one arrow is a counterexample.\n\nWhat is new and good: the observation that canonical actions of topological full groups of amenable ample groupoids are not strongly ergodic (with an appropriate hypothesis), leading to non-fullness of the crossed products for V and the Higman-Thompson groups, plus a non-embedding corollary. Lemma 3.1 (amenable groupoid gives amenable orbit equivalence relation) is straightforward and correct for the examples used. The proof via Rokhlin's lemma is the standard route, and for the intended examples—Cantor unit space, infinite orbits—it works.\n\nThe soft spot is the universally quantified statement. Schmidt's Rokhlin lemma does not produce a non-trivial asymptotically invariant sequence for finite equivalence relations; an amenable equivalence relation with finite classes can be strongly ergodic (the flip on a two-point space is the minimal example). So Theorem 3.2 needs a hypothesis excluding finite/essentially finite orbit structure—for instance, that G^(0) has no isolated points or that all orbits are infinite. Corollary 3.3 inherits the same problem: with the finite groupoid, take Γ = Z/2 acting on two points; it embeds into [[G]] and the action is strongly ergodic. The Section 2.3 remark that 'any amenable action is never strongly ergodic' is similarly overbroad.\n\nThe rest is in reasonable shape. The non-amenability argument via the Douglas–Nowak/Vaes–Wahl criterion is fine; the type III_{1/d} computation is standard. The author is honest that the almost invariant sequence is non-constructive. Typos are minor.\n\nWho should read it: people working on Thompson groups, topological full groups, and fullness of crossed products. With a corrected statement, it is a solid short note.\n\nRecommendation: send it to a referee, but the author must revise the statement and proof of Theorem 3.2 and Corollary 3.3. The specific Higman-Thompson and Brin-Thompson conclusions are unaffected and worth citing once the fix is in.","headline":"False as stated but probably fixable; the real Thompson-group results look right.","tokens_in":10082,"tokens_out":8554,"would_cite":false,"duration_ms":69415,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A40","20F38","20L05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the canonical actions of the Thompson group and its generalizations are never strongly ergodic, yielding non-full type III crossed-product factors and a non-embedding theorem.","keywords":["Strong ergodicity","Thompson groups","Topological full groups","Ample groupoids","Amenable groupoids","Crossed product von Neumann algebras","Type III factors","Cantor set"],"falsifier":"Search for a topologically principal, amenable, ample groupoid G and a quasi-invariant probability measure μ on G^(0) such that the canonical action of [[G]] is strongly ergodic; the theorem forbids any such example. Concretely, one can test the Higman–Thompson group V_{3,3} on the Cantor set: prove that some almost invariant sequence of measurable sets is non-trivial (as the theorem requires) or find a contradiction showing every such sequence is trivial.","tokens_in":8901,"feed_emoji":"🔁","tokens_out":12930,"duration_ms":95007,"temperature":0.7,"pith_summary":"The paper establishes that the canonical action of the Thompson group V on the Cantor set — and, much more generally, of the topological full group [[G]] of any topologically principal, amenable, ample groupoid G on its unit space — is not strongly ergodic for any quasi-invariant probability measure. The key step is showing that the orbit equivalence relation of such an amenable groupoid is Borel amenable, which yields a non-trivial almost invariant sequence for [[G]]. Because full crossed-product von Neumann algebras force strong ergodicity, this implies the associated group-measure-space algebras cannot be full. For the Higman–Thompson groups V_{d,k}, the paper combines this with a non-amenability argument and a type computation to conclude that L∞(X_{d,k}) ⋊ V_{d,k} is a non-amenable, non-full factor of type III_{1/d}; for the classical Thompson group V this is a non-amenable, non-full type III_{1/2} factor. It also yields the non-embedding of strongly ergodic actions, in particular free-group Bernoulli shifts, into these canonical actions.","feed_headline":"Thompson actions on the Cantor set are never strongly ergodic","feed_subtitle":"That means the associated von Neumann algebras are not full, ruling out a family of type III factors.","key_machinery":"The central object is the canonical action of the topological full group [[G]] — the group of compact open full bisections of an ample groupoid G — on the unit space G^(0). The load-bearing mechanism is the transfer of amenability from the groupoid G to its associated equivalence relation R_G: a net of probability measures on the source fibers, asymptotically invariant under G, is summed over each equivalence class to produce a net of probability measures asymptotically invariant under R_G, proving R_G is Borel amenable. Then the standard almost-invariant-sequence property of amenable equivalence relations yields non-trivial sets A_n with μ(A_n △ sA_n)→0 for all s in [[G]]. For the Higman–Th","core_discovery":"The central claim is that the canonical action [[G]]↷G^(0) of the topological full group of a topologically principal, amenable, ample groupoid is never strongly ergodic, regardless of the quasi-invariant probability measure on the unit space. The proof's engine is the transfer of amenability from the etale groupoid G to the associated orbit equivalence relation R_G = {(r(g), s(g)) : g∈G}: the fiber-wise probability measures witnessing amenability are pushed forward to equivalence classes, making R_G Borel amenable. Since [[G]] is contained in the full group of R_G, a non-trivial almost invariant sequence for R_G — obtained from the amenability of equivalence relations — is also almost invar","pith_inferences":["Because the proof is non-constructive, no explicit almost invariant sequence is given even for the Thompson group V; constructing one on the tail equivalence relation might reveal whether the sequence can be chosen to be clopen, giving a combinatorial obstruction visible in the Cantor set.","The non-embedding corollary suggests a general rigidity principle: topological full groups of amenable groupoids are poorly suited to hosting strongly ergodic actions of their subgroups; one could test whether this persists for actions that are only asymptotically strongly ergodic in a weaker sense.","The non-amenable, non-full type III_{1/d} factors exhibited here could serve as concrete test cases for questions about central sequences and property Gamma in type III settings, where few explicit examples are known."],"forward_implications":["The canonical actions of the Thompson group V, the Higman–Thompson groups V_{d,k}, and the Brin–Thompson groups are all non-strongly-ergodic, so their group-measure-space von Neumann algebras are not full factors.","For any topologically principal, amenable, ample groupoid G, no subgroup of [[G]] can realize a strongly ergodic action on G^(0); in particular, no free-group Bernoulli shift can be embedded as a restricted action.","The crossed products L∞(X_{d,k}) ⋊ V_{d,k} are non-amenable, non-full factors of type III_{1/d}; for the classical Thompson group V, the algebra L∞(X_2) ⋊ V is a non-amenable, non-full type III_{1/2} factor.","Since strong ergodicity is impossible, fullness of these crossed products cannot be obtained from central-sequence arguments based on the action; any fullness, if it exists, must come from a different mechanism."],"fun_headline_variants":["Thompson groups: no strong ergodicity on Cantor set","Thompson groups fail strong ergodicity on Cantor set","No strong ergodicity for Thompson groups on Cantor set","Thompson actions on Cantor set: never strongly ergodic","Thompson group actions: not strongly ergodic on Cantor set"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes that for every amenable etale groupoid the associated orbit equivalence relation is Borel amenable; if that transfer step (Lemma 3.1) were to fail for some topologically principal, ample groupoid, the existence of the non-trivial almost invariant sequence would no longer be guaranteed and the main theorem would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Thompson groups: no strong ergodicity on Cantor set","Thompson groups fail strong ergodicity on Cantor set","No strong ergodicity for Thompson groups on Cantor set","Thompson actions on Cantor set: never strongly ergodic","Thompson group actions: not strongly ergodic on Cantor set"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00049,"raw_usage":{"total_tokens":2155,"prompt_tokens":560,"completion_tokens":1595,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":304,"completion_tokens_details":{"reasoning_tokens":1511}},"tokens_in":304,"tokens_out":1595,"duration_ms":9765,"temperature":1.0,"reasoning_tokens":1511,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T09:48:54.758780+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a topologically principal, amenable, ample groupoid G and a quasi-invariant probability measure μ on G^(0) such that the canonical action of [[G]] is strongly ergodic; the theorem forbids any such example. Concretely, one can test the Higman–Thompson group V_{3,3} on the Cantor set: prove that some almost invariant sequence of measurable sets is non-trivial (as the theorem requires) or find a contradiction showing every such sequence is trivial.","supporting_citations":[],"review_version":1}